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Register a new userDesign and Stability Analysis of an Electromechanical Model for Nerves
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Added by
Francois M. Moukam Kakmeni
Publication date
2021
fields
Physics
and research's language is
English
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In this work, we propose and study the stability of nerve impulse propagation as electrical and mechanical signals through linear approximation. We present a potential energy stored in the biomembrane due to the deformation, bending, and stretching as the action potential propagates in the nerve fibre. From the potential energy, we derive electromechanical coupling forces and an attempt is made to unify the two models to account for both the electrical and mechanical activities of nerve signal propagation by introducing the electromechanical coupling forces. We examine the stability of the equilibrium states of the electromechanical model for nerves through the Routh Hurwitz stability criteria. Finally, we present results of the numerical simulations of the electromechanical model for nerves through Runge Kutta method of order four.
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In this work, we consider the electromechanical density pulse as a coupled solitary waves represented by a longitudinal compression wave and an out-of-plane transversal wave (i.e., perpendicular to the membrane surface). We analyzed using, the variational approach, the characteristics of the coupled solitary waves in the presence of damping within the framework of coupled nonlinear Burger-Korteweg-de Vries-Benjamin-Bona-Mahony (BKdV-BBM) equation. It is shown that, the inertia parameter increases the stability of coupled solitary waves while the damping parameter decreases it. Moreover, the presence of damping term induces a discontinuity of stable regions in the inertia-speed parameter space, appearing in he form of an island of points. Bell shape and solitary-shock like wave profiles were obtained by varying the propagation speed and their linear stability spectrum computed. It is shown that bell shape solitary wave exhibit bound state eigenvalue spectrum, therefore stable. On the other hand, the solitary-shock like wave profiles exhibit unbound state eigenvalue spectrum and are therefore generally unstable.
Localized spot patterns, where one or more solution components concentrates at certain points in the domain, are a common class of localized pattern for reaction-diffusion systems, and they arise in a wide range of modeling scenarios. In an arbitrary bounded 3-D domain, the existence, linear stability, and slow dynamics of localized multi-spot patterns is analyzed for the well-known singularly perturbed Gierer-Meinhardt (GM) activator-inhibitor system in the limit of a small activator diffusivity $varepsilon^2ll 1$. Our main focus is to classify the different types of multi-spot patterns, and predict their linear stability properties, for different asymptotic ranges of the inhibitor diffusivity $D$. For the range $D={mathcal O}(varepsilon^{-1})gg 1$, although both symmetric and asymmetric quasi-equilibrium spot patterns can be constructed, the asymmetric patterns are shown to be always unstable. On this range of $D$, it is shown that symmetric spot patterns can undergo either competition instabilities or a Hopf bifurcation, leading to spot annihilation or temporal spot amplitude oscillations, respectively. For $D={mathcal O}(1)$, only symmetric spot quasi-equilibria exist and they are linearly stable on ${mathcal O}(1)$ time intervals. On this range, it is shown that the spot locations evolve slowly on an ${mathcal O}(varepsilon^{-3})$ time scale towards their equilibrium locations according to an ODE gradient flow, which is determined by a discrete energy involving the reduced-wave Greens function. The central role of the far-field behavior of a certain core problem, which characterizes the profile of a localized spot, for the construction of quasi-equilibria in the $D={mathcal O}(1)$ and $D={mathcal O}(varepsilon^{-1})$ regimes, and in establishing some of their linear stability properties, is emphasized.
We present a stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs in whispering gallery mode resonators pumped in the anomalous dispersion regime. This article is the second part of a research work whose first part was devoted to the regime of normal dispersion, and was presented in ref. cite{Part_I}. The case of anomalous dispersion is indeed the most interesting from the theoretical point of view, because of the considerable variety of dynamical behaviors that can be observed. From a technological point of view, it is also the most relevant because it corresponds to the regime where Kerr combs are predominantly generated, studied, and used for different applications. In this article, we analyze the connection between the spatial patterns and the bifurcation structure of the eigenvalues associated to the various equilibria of the system. The bifurcation map evidences a considerable richness from a dynamical standpoint. We study in detail the emergence of super- and sub-critical Turing patterns in the system. We determine the areas were bright isolated cavity solitons emerge, and we show that soliton molecules can emerge as well. Very complex temporal patterns can actually be observed in the system, where solitons (or soliton complexes) co-exist with or without mutual interactions. Our investigations also unveil the mechanism leading to the phenomenon of breathing solitons. Two routes to chaos in the system are identified, namely a route via the so called secondary combs, and another via soliton breathers. The Kerr combs corresponding to all these temporal patterns are analyzed in detail, and a discussion is led about the possibility to gain synthetic comprehension of the observed spectra out of the dynamical complexity of the system.
We consider a bulk-membrane-coupled partial differential equation in which a single diffusion equation posed within the unit ball is coupled to a two-component reaction diffusion equation posed on the bounding unit sphere through a linear Robin boundary condition. Specifically, within the bulk we consider a process of linear diffusion with point-source generation for a bulk-bound activator. On the bounding surface we consider the classical two-component Brusselator model where the feed term is replaced by the restriction of the bulk-bound activator to the membrane. By considering the singularly perturbed limit of a small diffusivity ratio between the membrane-bound activator and inhibitor species, we use formal asymptotic expansions to construct strongly localized quasi-equilibrium spot solutions and study their linear stability. Our analysis reveals that bulk-membrane-coupling can restrict the existence of localized spot solutions through a recirculation mechanism. In addition we derive stability thresholds that illustrate the effect of coupling on both competition and splitting instabilities. Finally, we use higher-order matched asymptotic expansions to derive a system of differential algebraic equations that describe the slow motion of spots. The potential for new coupling induced dynamical behaviour is illustrated by considering examples of one-, two-, and three-spot solutions.
A dynamical theory of geophysical precipitation pattern formation is presented and applied to irreversible calcium carbonate (travertine) deposition. Specific systems studied here are the terraces and domes observed at geothermal hot springs, such as those at Yellowstone National Park, and speleothems, particularly stalactites and stalagmites. The theory couples the precipitation front dynamics with shallow water flow, including corrections for turbulent drag and curvature effects. In the absence of capillarity and with a laminar flow profile, the theory predicts a one-parameter family of steady state solutions to the moving boundary problem describing the precipitation front. These shapes match well the measured shapes near the vent at the top of observed travertine domes. Closer to the base of the dome, the solutions deviate from observations, and circular symmetry is broken by a fluting pattern, which we show is associated with capillary forces causing thin film break-up. We relate our model to that recently proposed for stalactite growth, and calculate the linear stability spectrum of both travertine domes and stalactites. Lastly, we apply the theory to the problem of precipitation pattern formation arising from turbulent flow down an inclined plane, and identify a linear instability that underlies scale-invariant travertine terrace formation at geothermal hot springs.
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